Thank you for that kind introduction and it's great to be here. You all for coming this evening?
I'm going to start by, as Eileen says, winding the clock back a bit and in fact, start by referring to what was an extremely what has proved to be an extremely influential popular science book, which was published in the second half of the 1980s by James Gleick, because it was this book that coined this iconic phrase, The Butterfly Effect, which I think has pretty much gone into everyday language and popular culture. In fact, the very first chapter of the book is called The Butterfly Effect.
And in this chapter, Glick describes the work of MIT meteorologist Ed Lorenz, whose picture is up there, and a paper that he wrote back in 1963, which was pretty much ignored, I would say, for ten years or so. It was published in what most scientists think of as an obscure metrological journal. Turns out, actually it's the premier meteorological journal in the field.
Anyway, there we go. But what Gleick describes, and I'm going to talk about this at some length in the talk, is how Lorenz came to discover three very simple equations which had no determinism or no randomness. Everything was precisely defined. The three equations coupled together in a way which I'll tell you.
And it had the property that it was almost impossible to predict the future of these equations, in the sense that starting with just a tiny, tiny, tiny perturbation, let's say, to the initial state, the solution developed quite differently. So let me actually illustrate that right at the beginning with this animation. So we're going to look at the evolution in time of what actually are. Two, it's not apparent until about now that there are actually two different solutions of these equations.
Starting as I say, from almost but not quite identical initial conditions. And after a while, although they track together the red and the blue for a while, they sort of correlate. And so knowing one of the solutions, if you like, tells you almost nothing about the second solution after a certain time.
Now, Glik then went on to say that this model of Lorenz basically formed a, you know, a hard quantitative bedrock for his notion that he'd been that he Lawrence had been talking about for some years of how the flap of a butterfly wings could go, let's say, in Brazil, could cause a tornado, let's say, in Texas. Now, this is where the history actually becomes slightly confused. And one of the. So I've got two parts to my talk.
One is to try to review what is pretty standard stuff, which is how Lorenz came to arrive at these equations and some of the amazing insights he had, incidentally, in linking these very simple, essentially equations that Isaac Newton would have understood very well with modern 20th century fractal geometry, a sort of concept that Newton would have found quite alien. So there's enormous insights that Lorenz had in deriving his 1963 model.
But I also want to say, and this is something which I'm sure most most of you will not know is that actually Glick got it completely wrong about attributing this 63 model to the idea that Lorenz had in when you referred to the butterfly effect. So unfortunately, the butterfly effect has actually been misnamed. And I'm going to try to explain why that is.
So the term the term well, the term the butterfly effect comes from Glick's book, but essentially refers to this paper or this talk, let's say, that Lorenz gave in 1972 at the American Association for the Advancement of Science and the title. It was a kind of a sort of semi it was essentially a public talk about predictability of weather. And that was the title. Does the flap of a butterfly wings in Brazil set off a tornado in Texas?
So what I wanted to do in the sort of towards the end of the talk is actually to to say a bit more about precisely what Lorenz said in this 1972 meeting. And you will see that it's actually not referring to his 63 paper at all, but to something he published, in fact, in 1969. And that is much less well known, I would say, in the broad community.
And in fact, this problem that Lorenz Lorenz wrote about in the 69 paper and which he talked about in the 72 meeting, turns out to be one of the great unsolved or closely linked one of the great unsolved problems in 21st century mathematics. So it actually gets to the heart of something which is actually much more radical than kind of chaos theory, although you might think chaos theory is fairly radical, but this is even yet more radical. So so that's the outline of the talk.
You'll know some of it maybe, and I'll fill in some details, but towards the end I'll say stuff that I'm sure most of you don't know. And I hope to I'm not going to really change popular culture. It'll it will always be called the butterfly effect. But I'm going to call what I what what Lorenz really meant the real butterfly effect. So we'll come to that towards the end of the talk. Let me just say a little bit about the biography of Lorenz.
So here he is, is as a young man, grew up in New Hampshire, very interested in astronomy as a kid, apparently had a passing interest in weather, but it wasn't a particularly sort of passion of his. In fact, probably weather was a nuisance because it obliterated the stars when he wanted to go out and look at them. But he claims, you know, in in later years, he had he had somewhat of an interest in weather. But he went to university, he went to study mathematics.
And, in fact, he went to Harvard and went through his undergraduate degree and in fact, started a postgraduate degree in mathematics under the great George Birkhoff, where he was studying some topic in apparently remaining in geometry, nothing directly to do with chaos. And his plan was to carry on. That was going to be his Ph.D. in mathematics. But then the Second World War came and sort of upset the plan.
And I think by chance, Lorenz discovered an advertisement that the CIA, the Army or the or the Air Force, I think it was the army at the time. They were looking for weather forecasters. Maybe it's the Air Force because it was two planes and so on. So he thought, well, okay, well, I've have a thought back to his childhood, a little bit of interest in weather. So maybe I'll that will be a good sort of career for me during the war.
So he he joined up and, you know, he was sufficiently good that after he'd been through this course, they decided he was the right person to teach the next lot of weather forecasters to be weather forecasters. So he actually became a tutor, if you like, in weather forecasting for the for the Air Force. He eventually got sent out to Okinawa, I think, in the Pacific. So he actually was involved in some of the real action in the in the Pacific sector.
Came back after the war and thought about, okay, do I restart my career in mathematics and decided, actually, you take quite an interest in a real interest in Weber. And so instead he enrolled at MIT just up the road to do a Ph.D. in meteorology. And the topic he was essentially given was trying to solve this equation. So it doesn't matter if you don't know this equation, but this is one of the archetypal equations in fluid mechanics.
It's essentially Newton's second law of motion, you know, suit and circular motion force. It was mass times acceleration. This is well, it's actually the other way around. It sets knots times acceleration equals force, but written for a fluid, which could be the atmosphere or it could be the oceans, or it could be a laboratory fluid, a fluid that potentially has many, many, many scales of motion.
And it is actually a remarkable thing that if you count, remember, the last time I counted the number of symbols, it was around 20 something, 22 or so.
Mathematical symbols is always strikes me as a wonderful thing that with just 22 mathematical symbols, you can describe the dynamics of every scale of motion in the atmosphere from the very largest jet streams, you know, which extends thousands of kilometres in length in the upper atmosphere, you know, right down to clouds, right down to the little eddies coming out of my mouth as I speak. This is all described by this one set of equations.
So this is Lorenz's job to look at. How do you know the problem is? So, okay, so this is this is this equation can be likened to very much a work of art. It is a work of art. But instead of likening it to a Renoir here, I'm going to liken it to a Russian doll. This is actually a very special Russian doll. Russian dolls unpack into small Russian dolls. This one unpacks into yet smaller Russian dolls and indeed yet smaller Russian dolls, and indeed yet smaller Russian dolls.
And indeed, yet smaller Russian dolls. Dot, dot, dot. And in the same way, if you actually want to solve this equation, you have to, as it were, unpack it also into actually what turned out to be billions of individual equations.
This is actually what makes weather forecasting one of the things that makes it so difficult, because you need to unpack it into the equations which describe the big jet stream, which describes low pressure systems, which describe clouds, which describe some cloud turbulence and all that stuff. So the way to solve these equations, which is what Lorenz did, is, is to look at truncating these equations.
Oh, sorry. Before I do that, yes. Let's let's say that these Russian dolls can be likened to, you know, the worlds or the eddies in a turbulent fluid. And in fact, it's it's probably appropriate at this stage to refer to the little piece of doggerel that was written by one of the founding. Well, one of the real pioneers of turbulence theory and also, incidentally, a pioneer of of weather forecasting. Lewis Fry Richardson, sometime in the beginning of the century.
Anyway, this famous little poem, big whirls of little whirls that feed on their velocity and little wells have lesser wells and so on to viscosity. Okay. What a great poet. But still, it makes the point that in a turbulent system, you have these many, many, many scales. So these scales, the big wells or the big Russian doll select little wells, the of the smaller Russian dolls and so on.
And the fact that this notion of feeding on their velocity captures this idea that, as it were, the Russian dolls can sort of bash into each other and transfer energy from one one Russian doll to another. And this is manifest mathematically in the fact that the equation here this you is the fluid velocity, and it sort of multiplies itself and that makes the system nonlinear and not linearity is the thing that allows energy to move up and down the scales.
And that's one of the complications of things. So Lorenz's PhD was actually about trying to whips was actually if I could go back, was trying to solve these equations in a sort of simplified way by getting rid of some of the scales and just treating maybe the system in a simplified, approximate way. And he came up with some what are called time stepping schemes and other types of what are called numerical schemes,
which actually even to this day are still used. Anyway, the study was, I would say, moderately successful. It wasn't like I set the world on fire, but it was moderately successful. And he found that within a few years of his PhD, he'd been invited back to MIT as a proper member of faculty. But there was a slight rider on this appointment because he had to be in charge of a group that was whose kind of principal research activity was doing long range weather forecasting.
So long range means, you know, a month or two ahead, whereas the people that were trying to kind of approach weather forecasting from the from the point of view of solving these navier-stokes equations were only thinking about, you know, maybe 12 hours ahead at the moment or maybe a day ahead at most, but just a very short time. So how would people even think about doing long range forecasting? So this was actually done by a completely different method to solving these equations.
It was just done with the statistics. So the idea is you have a big pile of weather maps which let's say we do this today. We got a big part of weather maps from today going right back to whenever you like, 1800 for the sake of argument. So we're in what have we our may. So let's suppose our task is to forecast what the weather's going to be like a month from now. So the monthly forecast for June or July or something.
So the idea that this group had and incidentally, he got a lot of this group got a lot of support from the Statistics Department at MIT. So this this this had some pretty high level support was you just go back in time and find a weather map that from the past that looks like today. So you go back and find, let's say 1961, May 1961, the weather looked very similar to May 2017.
Then what you do, you predict for June, July 2017, what the weather was in May and June of whatever I said, 1961, it's kind of method of analogues. So Lawrence got put in charge of this team, and his first reaction was, Hmm, this didn't sound right to me. You know, what's the scientific basis for this? This idea of just so you can find an analogue? Because it seemed to suggest that the weather was somehow very periodic.
You know, if it if it was safe, it was like it is in May 61, that necessarily what we see in the next couple of months will be what happened in 1961. And his kind of intuition was that the weather isn't periodic like this. It seems to be irregular. And apparently had long battles with people not only in this group, but again with the statistics, people that I met. And he said, you know, I've got to try and sort this out. I'm going to try to prove that this really this idea isn't going to work.
How am I going to do it? So that's when he went back to his truncated navier-stokes equations. So maybe I can prove that this property of of periodicity just doesn't happen. If I take a semi realistic model and use I mean, computer technology was, was starting to in the fifties, computer technology was starting to arise so I could solve these equations on a computer and just show them that this won't work. So he set about, you know, taking actual weather type of equations.
And for a long time, he worked with many Russian dolls of the eight. So actually, he worked for a long time with 12, you know, a 12 component. So truncating the equation down to 12 compares. But, you know, the computers just weren't up to the job. And he found this was really hard work trying to do anything. It just took too long to get anywhere.
And he had a real breakthrough when he talked with a colleague of his from a nearby university who had been studying not so much weather, but what's called convection in a in a fluid where you heat a fluid from below and look at the circulations that develop. And this guy actually said that he had a seven component model that seemed to seem to have this property of non periodicity.
Maybe Lorenz could look at that and Lorenz did in fact move on to that problem and he realised within this subset of seven equations there was actually a three. So he realised when he was looking at these non periodic solutions that four of these components almost went to zero. So we kind of had the intuition that there would be a three component. He had a three component subset that. That had the required property. And this is what led him to these iconic equations, to Lorenz 63 equations.
So this is just a three Russian old truncation of a laboratory fluid. So you've got three variables X, Y and Z. So these are not space. These sort of describe a type of circulation. It doesn't really matter. This this equation has been truncated so much, it now sort of likes to lose contact a bit with the real world. So it's kind of an idealisation of the real world. So X, Y and Z, just variables thinking the variables.
And in the three equations, the left hand side, the x by d t d y by d t d set by d to use the time rate of change of X, Y, and Z. And it's given by these terms on the right hand side. The numbers, ten and 28 are not. They don't have to be precisely those. These are the numbers that Lawrence used. But you can, you know, choose numbers nearby and you get the same behaviour.
And you notice that in the right hand side you get these terms multiplied together, X and Z multiplied together, X and Y multiplied together. So it does retain that notion of nonlinearity, which as we come to, is all important. And when Lawrence integrated these equations. So now we're looking at the the X component variable as a function of time. You can see it's, it's, it doesn't have any it doesn't you know, it, it looks very irregular.
There are periods when it seems to be up in one state, but then it kind of jumps down or periods when it's down here and then it jumps up. So this is exactly what he was looking for to try to kind of prove this counterexample to the statistical model. But the more Lorenz looked at this, the realised he didn't really understand what was going on, what on earth, you know, what is causing this behaviour. And he had the real insight into plotting it in a different way.
So instead of taking one variable and plotting it again, this time you take all three variables and imagine a three dimensional space span by those three variables and then let time kind of represent be represented by the length along a trajectory. So this represents some fairly random starting condition in this three dimensional space of X, Y and Z. So you give it anything you like and then let it go with his computer. And what he found was that the trajectory start to fall.
Sort of strange shape. Now, the first thing is you can see it's got this these two kind of wings, if you like. And sometimes people call this a it looks like a butterfly, but that's entirely coincidental. But it has these two kind of wings which which in a sense describe those that two that kind of regime behaviour, it tends to be the variable tends to be up here or down here. Well those if you like up here, up here is, is going around here and down here goes round here.
So that was kind of that was an interesting thing. But then Laurence sort of said, well, what exactly is this thing that this this trajectory is kind of oscillating on? What is what is the geometric object behind this? And for a while he thought maybe it's some kind of surface. These two lobes are lying on the surface and somehow the surface is just sort of glued together. But then he realised that couldn't be the case because. Then the trajectories would have to kind of cross over each other.
It just didn't work. And it was he agonised about this for by the way, I just want to talk about that. I've got a. A nice animation of the of how the actual state is not going. They just don't work. Yeah. It doesn't seem to work anymore. Okay. Not to worry. This would have been an animation showing a the state going round in a very irregular way around what is going to go. Okay. And you've got different perspectives. That's now the Y in the Z. I think you had the. The Z and the extraction.
That's the name of the X and Y direction. Then it'll rotate round to the. So Lawrence kept asking, So what on earth is this geometry? What is it? What is it? Actually, what? What how can I describe this in a kind of mathematical way? And that's when you realise this has to be some kind of fractal. Now maybe these days we're kind of blasé about fractals, we see them everywhere, but just kind of caution went back to the 1960s.
He's dealing with equations which as I say, Newton would have had no trouble understanding and suddenly how this fractal geometry kind of comes out. So what I want to do is spend a few minutes trying to get this insight into why why a fractal? And to do that, I'm actually going to use a slightly simpler dynamical system.
It's just easier to explain. And this actually was was was derived, as you can see, 13 years later, by wrestler in a way, as a way of of kind of really trying to describe this phenomenon of chaos as as simply as possible. And again, it's a three component system of differential equation. So the DOT stands for D by d t, so divided you've actually divided by two. You've said but slightly different equations. But again, notice the non linearity where pointer is.
Okay. So what I want to do is take you through the rustler a tractor and try to understand why that is fractal. So I'm going to use some nice pictures from a book by Abraham Henshaw in 1984. So first let me start with so what I want to introduce are three essential components for this type of chaos. And the first two components are illustrated here. And this is generic to all chaotic systems. So imagine you take a little area here.
This is in this space of states, state space, X, Y or Z, whatever. And we're going to look at, let's say, two, two points on the edge here and let them evolve in time. Now, what this what what these lines are illustrating is the fact that one of the essential ingredients of chaos is the notion of instability, that the two points that are initially close start to diverge in general exponentially from each other.
And that's kind of illustrated, as I say, here. So this this point goes off in this direction, whereas this one goes off in this direction. So this direction here is a kind of direction of instability where initially close trajectories are diverging apart. So that's one key ingredient. The second key ingredient is actually looking at this in the sort of transverse direction.
So looking at points, let's say, which start on the other edge of the square, because the second feature of chaos is that these or this type of chaos at least, is that these trajectories in this direction are actually converging together. So they're getting closer. Now, the key point is that overall. Average over the whole system in all time. It's actually the converging direct directions win over the expanding directions. So that if I was to measure the area of this.
So here's if I take this area at initial as imagine this is at some initial time and then that evolves into an area. Imagine looking at this object from the far side. Then there'd be an area here which was stretched out in this direction but compressed in the transverse direction. Then on average, the area will be smaller afterwards than it was before. So this overall shrinkage of area. So these are two key ingredients for chaos.
So this just shows the same thing again, except they've just drawn or they've drawn this the transverse, if you like, direction as a, as a rather small, um, just much smaller. Okay. So that's fine. Now we have a problem. I mean, if, as I say, these if you just imagine exponentials of divergence, then this would just carry on. They would diverge as far apart as you like, go off to infinity, as it were.
So this can't possibly explain the fact that we have a a kind of a geometry which seems to sit in a in a finite, compact region of state space. So how do we stop the trajectories going off to infinity? Well, this is where the third ingredient of nonlinearity comes in. And in the case of the Russell Attractor, it's rather simple. The nonlinearity just folds one of this. The surface over like this. So and so.
Just take one hand and just fold it over. So let's have a look at that in a bit more detail. So let's let's take our. Let's talk a little surface. Here are two points, two trajectories. And they kind of evolve and the system becomes nonlinear and the surface folds over like this. So that one trajectory has gone to the top part and the other trajectory to the to the bottom part there. And it's kind of folded over in half.
Now, if you remember what I said, if you try to let's try to kind of compress that down into the same cross-sectional area as we started with. Then what I said, if you if you recall, is that this is the combined area or the total area of this top part. Plus the bottom part is actually less. It's smaller because of the overall effect of this contracting this. This is the effect of dissipation, effectively. This total area is smaller than the the one I started with.
So if I try to compress the whole thing into the same area, there must be a gap between these the top sheet, if you like, and the bottom sheet. This is a really crucially important point. There's a gap. There has to be a gap. They don't kind of completely because you've got some compression effectively. So. All right. So let's take this thing again. You can imagine this is just like here, but with the gap. But I've sort of blown this up a bit.
And so this kind of comes round as two sheets and we'll go we'll go through the whole exercise again. Fold it over. And now you see what was two sheets have become force sheets. Now this gap. Has now kind of formed two small gaps, if you like, here and here. And a bigger gap has appeared in between them for the same reason that I said before.
And now we do it again. We got now these four sheets we folded over and now we've got eight sheets with a big gap, two small gaps and four really small gaps, which was the original gap done, done twice over. Now, if we keep doing this over and over and over and over and over and over and over and over, what are we going to end up with? Okay. Well, one way to to try to understand that is to take a cross-section through this this structure here.
After, you know, a very large number of of these types of iterations, this is sometimes actually called a Lorenz section. The whole thing is sometimes called a Poincaré section. And this kind of transverse through the trajectory is called a Lorenz section. So what is this? Well, this was this structure was discovered some years earlier, in fact, in the late 19th century, early 20th century by the great mathematician George Cantor.
And his reason for for this he invented this set called the Cantor set was really nothing to do with chaos per say, but really to try to understand a bit more about the nature of numbers. And this is an interesting set is an interesting example of a of a set which has an uncountable infinite number of points. But but the points actually take up no volume at all. No volume at all in the on the on the line.
So the Cantor set is is constructed by starting with just, let's say, the line between zero and one and throw away the middle third. So you got two pieces and then throw away the middle third of the remaining two pieces and then throw away the middle third of the remaining four pieces and continue, continue. And then just take the intersection of all of these iterates and you're left with this set. Now, so this big the big gap. Is is if you like.
Is this part of the of the rustler a attractor or it's a gap in the rustler tractor. And then the next iterate are these two smaller gaps and so on and so forth. So we have this. So this is where fractals basically come from. The three ingredients, instability, dissipation, non linearity. And this is what Lawrence said in his 63 paper. We see that each surface is really a pair of surfaces, so that where they're they appear to merge. They're really four surfaces.
Continuing this process for another circuit, we see they're eight processes and we finally conclude there's an infinite complex of surfaces, each extremely close to one another or other, the two merging surfaces, I think. Ian Stewart in his popular science book, Does God Play Dice? Absolutely Nails the reaction, I think of a lot of mathematicians when they read this paper and he said, when I read Lawrence's words, I get a prickling at the back of my neck and my hair stands on end.
He knew 34 years ago. He knew. And when I look more closely, I'm even more impressed in a mere 12 pages. Lawrence anticipated several major ideas of nonlinear dynamics before it became fashionable.
So if I had to say, you know, one thing that Lawrence for me absolutely characterises the genius of Lawrence, it's actually not so much the, you know, this divergence of trajectories, but the realisation of this fractal geometry, which underpins these sets of differential equations, which by all accounts should look very smooth and unexceptional.
Unexceptional. So with that in mind, I am going to make the claim that in some sense Lawrence provides a bridge really between the classical physics and classical mathematics of Newton and some of the real key ideas in 20th century math. So we've already mentioned Cantor. This is Kurt Gödel and this is Alan Turing.
And I'm sure people know that some. They really revolutionised our ideas in mathematics by bi by raising the notion that there are there are things which are true in mathematics but not provably true or in Turing's language propositions, which are kind of undecidable algorithmically, no matter how complex or your computation might be, you just cannot establish the truth of of certain propositions.
And it turns out that many of the properties of these fractal geometries are quite isa morphic, quite similar to the types of propositions of good Alan Turing that are undecidable. In fact, the papers written on this showing this in a formal way. So there is this bridge between, as I say, the calculus of Newton, which, as I say, Newton would look at these equations, say, yes, I take that, I understand that.
And but through this remarkable geometry that they generate, we have these links to some of the most profound things in in mathematics. Now, I've included Andrew Wiles here because there's lots of other interesting things you can ask.
For example, suppose I want to have a type of arithmetic where I add numbers on on this or on this fractal set, on this cantor set, let's say on this transverse cantor set of multiply numbers such that the sum of the numbers or the product of the numbers also lies on the cantor set. Now, can we do arithmetic on the Cantor sets? It turns out you need a special type of number system to do that called padding numbers.
So I think numbers are a very rigorous branch of pure mathematics that has a very close link to fractal geometry. And as I say, they they respect the sort of geometric constraints of fractals in doing algebra. And it turns out that a lot of the deep theory a number theorems from poetic numbers systems were very important in Andrew's celebrated proof of Fermat's Last Theorem.
One of the things which really intrigues me as a physicist is whether there are links between these ideas and quantum physics. So people may recognise Erwin Schrödinger and Heisenberg, Paul Dirac and another very celebrated Oxford mathematician, Roger Penrose. Suddenly three of these people, Schrödinger, Dirac and Penrose, I'm not so sure about Heisenberg, but the three of them were very uncomfortable with quantum theory.
It works and it does work to this day, but the foundations are very the why this keeps jumping. The foundations are very kind of bizarre and difficult to understand, and I think all three of them felt that there must be something deeper underpinning quantum theory, something much more deterministic.
But Roger Penrose in his books like The Emperor's New Mind, which I'm sure some of you will have read very much, emphasised that if we if there is some deterministic underpinning to quantum theory, it has to have some notion of non compute ability. This undesired ability has to feature somewhere deeply. Just an ordinary type of deterministic system won't do it.
So the reason I've included Roger here is because he has very much stressed the potential role of non compute ability in going deeper into fundamental physics. So again, perhaps these types of fractal objects play a role. I personally believe they do, but one has to say at the moment it's a matter for uh, for study and so on. So I'm not going to dwell upon this any more in the talk, but I'm happy to talk to people afterwards if they're interested.
Chaos theory, for sure, has revolutionised many different branches of science in physics, in biology, in engineering, in economics and social science. It's hard to think of any area where it's left untouched. Unfortunately, it's sometimes also abused. So I want to give an example of an abuse of chaos theory. And this is actually something that's close to my heart because I have to deal with these sorts of people quite a lot.
The climate, well, they don't like to be called deniers, climate sceptics. Now, one argument which is quite common actually goes like this You guys, okay, you might be able to forecast the weather tomorrow, but you're pretty hopeless. And even then, you sometimes get it wrong. And, you know, a couple of weeks ahead is pretty dodgy. A year ahead, no chance. So what wasn't on it? Why on earth should I believe what you say about the climate 100 years from now?
So this is this was this was brought up not so long ago in a review of a book in The Telegraph. Daily Telegraph. The game's up for climate change believers, the theory of global warming as a gigantic weather forecast, and therefore it can have a century or more and therefore can have no value as a prediction. Okay. So I just want to debunk that very briefly and then we'll move on. Because basically that is to misunderstand the problem of climate change.
What I've done here is to do a kind of climate change experiment in Lawrence world by adding an extra term to one of the equations. Now, just think of this term here. This is actually just going to be a number. But just think of this as a kind of a surrogate for doubling carbon dioxide concentrations in the atmosphere. So I'm a sort of applying an extra forcing term, if you like, to the equations. So how does that affect the Lorenz the Lorenz system?
So here's again the equation X component as a function of time before the system would oscillate roughly between these two kind of regimes, the two lobes of the butterfly with equal sort of frequency. So it is likely to be up here in one lobe of the butterfly, as it would be to be down here. What's happened with this additional term is that I've increase the likelihood of it being in the upper lobe and decrease the likelihood of it being in this lower lobe.
Now, the system is still chaotic because if I want to predict in detail at one of the trajectories, if I want to make a weather forecast, if you like, in this Lorenz world, it's still going to be predictable. It's still going to be subject to uncertainties in the initial conditions. But what is predictable is this kind of gross statistic that the probability. Of the state being up here has been affected in a very predictable way.
In fact, we can you know, we can actually plot this on in the Lawrence attractor world in the state space. So here's the original Lawrence Attractor in the new system, these two kind of what we call centroid. So right in pretty much the same place as they were before. But now the system spends a lot more time zipping around here than it does zipping around here. So we could view in this perspective, we could view climate change as a problem in geometry.
The question is how is the geometry of this attractor being affected by the addition of this term, which I've now set equal to ten? So it's important to understand this. So climate change, this is the problem of climate change. How does the whole climate attractor it, which basically means the statistics of weather change as we double carbon dioxide levels, which we surely will be sometime later this century or so over pre-industrial. How how how does that change the statistics of weather?
And that's not the same problem at all as completely different class of problem to saying how does a particular trajectory evolve on its on the climate attractor. Over some from some initial condition. Okay. Uh, let me just now move on to the sort of part of the talk I wanted to, uh, I wanted to get to, which is what did Laurence actually mean? By the butterfly effect. And you'll see that, in fact, he wasn't talking about his 63 paper at all.
He's talking about the fact that the weather is a multi scale system. So we have we have a you know, if you imagine a large low pressure system trundling across the Atlantic embedded in that low pressure system, there'll be, let's say, you know, thunderstorm clouds, big thunderstorm clouds where they match the some of the storm clouds, maybe 100 kilometres in scale, whereas the weather system itself is a thousand kilometres in scale.
Embedded in that thunderstorm cloud, there are small sub clouds, turbulent eddies, and within those sub cloud, turbulent eddies there, yet smaller turbulence eddies. So we're looking at this this Russian doll hierarchy again. Now, Laurence said, okay, let's suppose our problem is to predict this weather pattern, the large scale weather pattern. That's what we're interested in.
We might be able to measure, you know, the initial the starting conditions for this weather pattern very, very accurately. In fact, we can imagine almost perfectly for the sake of argument, measuring the initial conditions for that weather scale.
But that's not going to give us indefinite predictability because sooner or later the fact that we haven't been able to measure the the cloud scales or the sub cloud scale motions perfectly, that's going to catch up with us because these areas are going to kind of propagate up non-linear, only from the small scale to the larger scale to the very large scale. If you want to actually read this, he's Lawrence himself wrote a popular book called The Essence of Chaos.
And in the appendix, he actually describes what went into this talk in the triple OAS meeting in 1972, and he talks about errors in the course of scrapped structure. So that's a large scale weather pattern and then errors in these final structures, which are the clouds. And then in fact, the fact that errors in the final structure can start to induce errors in the course of structure.
What I've decided to do for this talk is actually not to go through in detail with what he wrote, but return to my Russian doll example and try to describe what he what his thinking was with that. So let's imagine the Russian doll, the big Russian doll is this big low pressure system trundling across the Atlantic. And that's the thing we want to predict. We want to predict that as far ahead as we possibly can. And let's imagine we've got some observing, some instrument which has no error at all.
I mean, infinitesimal, just quantum mechanics, just for the sake of argument, measures perfectly measures whatever its wants to do. Temperature, pressure and stuff like that. Now we have a whole load of these instruments and we're going to adopt them around the whole globe.
We've got enough of them. We can dump them around the whole globe in a regular network where the distance between the two, between any two of these measuring systems is sufficiently, let's say, small, that it can it can resolve, you know, this this low pressure system really, really well. So that the initial conditions for this low for the scale, this low pressure system, the scale of this low pressure system are known almost perfectly.
But the distance between these measuring objects is not sufficiently small that I can measure the initial conditions for all these smaller eddies, the smaller wells of Richardson. So I'm going to imagine that with that information. Let's say I can predict ahead five days. I mean, it's a reasonable sort of number. So what Lawrence said was, well, maybe we're not happy with that. How would we extend that predictions? Horizon We want to predict further.
Well, what's limiting us in this case is not our ability to measure this scale, because we've measured this perfectly. What's limiting it is the fact that we don't have any information about this scale. So we'll stick a whole load more of these instruments down, filling the gaps between the ones we had. So now we have enough resolution that we can measure the initial conditions of this scale here.
So the question is, how much is that going to bias? How much extra prediction skill is that going to give us? Now, Lawrence says the answer is not ten days, because these these things, the errors are likely to be growing faster for these smaller scale features. So, for example, you mentioned the cloud system, the error, the time it takes because the circulation patterns are more vigorous in a in a in an individual cloud.
If you have a flown through a cloud in an aeroplane, you know, speaking of thunderstorm cloud, there's a very vigorous circulation. So air is a growing much more rapidly. So let's I'm kind of simplifying the for the experts. I'm simplifying things a little bit just to make the point. So but nevertheless, the idea is, is qualitatively correct. So let's imagine for this second, you only get two and a half days of extra prediction skill.
So you've gone up now from five days to seven one half days. So Lauretta says, okay, that's fine, but let's, let's say we want to go further. So let's, let's have a whole load. More measurement systems are now fill in even the gaps in those gaps. So then now we can even measure this Russian doll initially perfectly and he says, okay, again, we'll only because these are growing yet faster, the errors are growing faster. We may only get an extra C I'm I'm halving the extra time each step.
So now we only get an extra day and a quarter. So you can see this continual investment in imperfect measurement measurement systems, which is driving the initial error closer and closer to zero, is actually buying us less and less prediction time. And what Lawrence says. Let's take this to the limit and imagine we've got an infinite number of measuring systems which can basically take us down to infinitesimally small scales.
He's saying that what this will bias in terms of prediction time is a series which if with these particular choices of doubling times, will actually converge to a finite number. Now, normally we have when we have series, we think of convergence as a good thing and divergence is a bad thing. But here, convergence is a bad thing because it's really limiting your prediction capability to some finite time, which in this simple calculation turns out to be ten days.
So this has a remarkable. If you think about it, this is a remarkable thing. If it's correct, it says. No matter how small my initial error is, I will never be able to predict more than ten days ahead. Now, just think that's completely different to the 1963 paper, because the 63 paper is about just these three equations, which, although it's very difficult to predict them, you can always in principle predict as far as ahead as you like.
Providing your initial error is sufficiently small. But here's a system where that's not true. And this is what it's about in his 69 paper, which appeared in a Swedish journal called Tell US. And if his 63 paper was, you know, is considered to be in an obscure journal, this is obscure squared, I'm afraid. I'm afraid so. Very, very not. Not really. Well read at all. But it's worth just if you don't mind just reading, I'll read out the the part of the abstract.
So it is because it's really important this he claims it is he proposed that it is proposed that certain formally deterministic fluid systems which possess many scales of motion. That's the key point. Many scales of motion, not the three rationales, but a kind of essentially an infinity of Russian goals, are observationally indistinguishable for in deterministic systems.
Specifically, that two states of the system, differing initially by a small observational error, will evolve into two states differing as greatly as randomly chosen states of the system with a within a finite time interval. In our calculation, it was ten days. Now, here's the real Ryder kicker, which cannot be lengthened by reducing the amplitude of the initial error. So this is very radical. So what I'm proposing to do here today is to illustrate so as to introduce you to two butterflies.
Now it's we're never going to stop the butterfly effect being used to describe low order chaos. But I'm going to call this the common butterfly effect. So here's a common blue on the common butterfly. This is the one that everybody thinks of when they talk about the butterfly effect is about sensitive dependence on initial conditions. So that means it's certainly difficult to predict the future, but it's not impossible in principle.
You can predict in something like a Lorenz 63 system, you can predict as far ahead as you like. Providing the initial error is small enough. On the other hand, I'm going to use this monarch butterfly. So this is a slight, a weak pun on the fact that this is monarch is a royal member of royalty and royal reality in Spanish or something is royal.
So the real butterfly effect, the royal butterfly effect, if you like, is the 69 paper which basically says that there are finite predictability horizons which cannot be extended by reducing uncertainty in initial conditions. So we know that this is this has been certainly well verified. So the common butterfly effect is been is kind of absolutely part of standard science. How about this real butterfly effect? Is this part of a of a kind of rigorous science?
Let's just put this into a slightly more mathematical phraseology, because what what we're saying here is if you take the initial conditions and change them as slightly as you like in a finite time, you'll you'll diverge, too, to find actually different solutions. Now in math, in mathematics, problems are referred to as well posed or ill posed according to certain properties that they have, apparently from a definition given originally by had them.
So problems are well posed if solutions exist, they're unique and the solutions depend continuously on the data. This means initial data in our case. Now, this continuously means if you change the initial conditions very, very slightly, you're just going to change the solution slightly. And that's true in the 63 chaos, but it's not true in the 69 case, in the 69 real chaos, real butterfly effect.
So if this effect is really true, it means that the navier-stokes initial value problem is not well posed in this sense. So now this has becomes a problem in mathematics. It's the navier-stokes problem for three dimensional, turbulent fluid. The initial value problem well posed. And this is why I say this is actually turns out to be one of the great unsolved problems in 21st century mathematics.
So as I'm sure many of you know, the Clay Mathematics Institute put out these kind of key problems, unsolved problems in mathematics, which include famous things like this is the beginning of this millennium, the Riemann hypothesis and so on. And here's the the Navier-stokes one. It's actually not framed in terms of this continuity property, but rather whether solutions exist and are unique.
And even that's not known. And we need to know that even before we can understand this property of continuity. So I think it's fascinating to me that what Lawrence speculated about in 69 and then this triple as talk in 72 actually refers to a very deep problem in maths.
So if I got just a few minutes to finish, okay, so I want to kind of get back to two more practical issues because a question you might ask is, is it really the case then that we're limited in weather prediction to ten days or something and things aren't quite as simple as that in practice because. Ten days may well be sort of roughly an average timescale that we can make sort of detailed predictions about the weather.
But what we rapidly find out is that there are many situations where we can make much longer range predictions, apparently with with quite good skill, but equally other situations where predictions do in fact go wrong within a couple of days. And that's very well illustrated going back to the 63 paper and actually looking at how errors, if you like, or uncertainty. So imagine this is an initial state with some kind of ball of uncertainty associated with it.
This actually shows that there are some parts of the Lorenz attractor that are extremely stable, so small perturbations hardly grow at all. So this is telling you there are some parts of the system which are actually very predictable, other parts of the situation, other parts of the attractor, where you start to explore the state space down here, where you do start to see unpredictability and some initial conditions where the uncertainty just explodes.
Now, those of you, most of you in the audience are not old enough. But I know there are a few people to remember this. But one of the most iconic weather events of the last maybe 300 years occurred almost 30 years ago, October 1987.
And this poor guy whose name is Michael Fish, although he's I have to say, he makes a great after dinner living out of speech, living from this famously made probably the worst weather forecast in history by saying there's no chance of a hurricane hitting the UK in the next a day or so. And indeed, a hurricane did did hit. And this is actually a great example of this very intermittent phenomenon of explosive unpredictability.
So what I'm going to do here is just animate two weather forecasts from two almost identical initial conditions. These are these are surface pressure maps, almost identical. Not quite identical. You can look in detail. You'll see differences. With just a couple of days before the storm hit. So just run this forward and they start off tracking each other pretty well. But then leading up to the fateful morning of the 16th, I think it was, you see that these are really quite different.
This would be a very kind of benign day, and this is a really intense vortex. Poor Michael Fish was given essentially this solution by the Met Office, but for a flap of a butterfly's wing, he would have been given this one and he would then have been a national hero instead. Actually, you shouldn't feel too bad about this because the Met Office did really well out of this poor forecast.
They said, Oh, well, our computers aren't big enough, you need to invest lots more money. And so I don't feel sorry for them. Okay. So the question is, how would a modern day Michael Fish deal with this situation? Because chaos is chaos. I mean, chaos in 1987 will still happen again. So this intermittent explosive unpredictability could still happen in 2017. So how would a modern day Michael Fish deal with this situation?
Well, these days, with much bigger computers and in particular with massively parallel computers, we can take advantage of that type of architecture to run weather forecasts in what I call ensemble mode. So we actually we actually run here 50 different forecasts. This would be done typically every day at places like the Met Office.
Some are actually everywhere around the globe. Now, you don't run a single weather forecast, you run an ensemble of them and very, very slightly the initial conditions and look to see ahead of time whether the forecasts are diverging and what you see in this case. So this is the 87 storm run retrospectively. It's run with a modern day ensemble system, modern day weather forecast model.
But but on this on this 30 year old case and what this shows at about a two and a half day range is a phenomenal spread in in solutions. So these are all pressure maps. So you see, you know, you can get pretty much everything from, you know, absolutely balmy, calm days to to horrendous weather, very, very wide divergence. Okay. So what would you do with this information? How would you sort of how would you convey this information to the public?
Well, one thing you can do is basically say, okay, how many of these members had hurricane force winds and count them? And if there were, say, 15 out of the 50 had hurricane force winds, you would say 59, 50, 30 out of 100. So you say is roughly a 30% chance of. Hurricane force winds, and that's actually pretty much what we see. So this is a a contour map showing the probability of hurricane force wind gusts on the 16th of October based on that ensemble forecast.
And these are blue lines here. And the English Channel are up around 30, 40%. Now. So what do you do with that information? That's up to you. That's not up to the meteorologist. That's up to you. If you've just bought a brand new yacht and you're not terribly good at sailing and you're thinking of crossing the channel,
maybe don't want to do it. If you just bought a brand new Lamborghini and it's sitting underneath a big oak tree, maybe you want to move it. 30 or 40% is probably a big enough probability that that would be a prudent thing to do. But it raises a very interesting and a whole topic for another talk some time about making decisions under uncertainty. But the main point is, and indeed much better decisions can be made once you have a quantified quantify uncertainty.
And that's precisely what this does. All right. So I think I'm finished. Sorry for going slightly over time. I'm going to be another shamed advertisement. If you're interested in reading any more about anything I've spoken about, I just want to highlight three papers which I can send you or you can get them on the on the Internet.
Most of this talk is actually based on a paper that I wrote with colleague Andrea stirring from the physics department and Gregory Sagan, who's here in the Maths Institute, about this butterfly real butterfly effect.
And we go into a lot more detail about the theory of the navier-stokes equations and things like that, but as well as the history behind the you know, what I spoke about, I also wrote a biographical memoir for at Lawrence, who died a few years ago, who was a former member of the Royal Society. So there's a lot of the biographical background to Lawrence in that which you can again get from the Royal Society and if you're interested in this area of this.
So what I find rather fascinating linkage between Lawrence and the girdle incompleteness theorems and some of the ideas of Roger Penrose on non compute mobility in physics. That's in a paper in contemporary physics published by the Institute of Physics, based on a lecture I gave in the physics department a few years back. Thank you for your attention.